the sum is sinθ1 + sin2θ2 + ...
Let S = sinθ1 + sin2θ2 + ...
and C = cosθ1 + cos2θ2 + ...
C+ iS = eiθ + e2iθ/2 + ...
= - log(1-eiθ)
= - log(1-cosθ- isinθ)
= loglzl + iarg(z)
So S = arg(z) = tan-1(-sinθ1-cosθ)
the sum is sinθ1 + sin2θ2 + ...
Let S = sinθ1 + sin2θ2 + ...
and C = cosθ1 + cos2θ2 + ...
C+ iS = eiθ + e2iθ/2 + ...
= - log(1-eiθ)
= - log(1-cosθ- isinθ)
= loglzl + iarg(z)
So S = arg(z) = tan-1(-sinθ1-cosθ)
adding to the soln
tan-1(-2sin(θ/2)cos(θ/2)2sin2(θ/2)
= tan-1(-cotθ/2)
= -tan-1(tan(pi/2-θ/2)
= θ/2 - pi/2